A phylogeny, or evolutionary tree, represents the evolutionary relationships among a set of organisms or groups of organisms, called taxa (singular: taxon). The tips of the tree represent groups of descendent taxa (often species) and the nodes on the tree represent the common ancestors of those descendants. Two descendents that split from the same node are called sister groups. In the tree below, species A & B are sister groups  they are each other's closest relatives.

Many phylogenies also include an outgroup  a taxon outside the group of interest. All the members of the group of interest are more closely related to each other than they are to the outgroup. Hence, the outgroup stems from the base of the tree. An outgroup can give you a sense of where on the bigger tree of life the main group of organisms falls. It is also useful when constructing evolutionary trees.

What's the difference between a phylogeny, an evolutionary tree, a phylogenetic tree, and a cladogram?

For general purposes, not much. This site, along with many biologists, use these terms interchangeably  all of them essentially mean a tree structure that represents the evolutionary relationships within a group of organisms. The context in which the term is used will tell you more details about the representation (e.g., whether the tree's branch lengths represent nothing at all, genetic differences, or time; whether the phylogeny represents a reconstructed hypothesis about the history or the organisms or an actual record of that history; etc.) However, some biologists do use these words in more specific ways. To some biologists, use of the term "cladogram" emphasizes that the diagram represents a hypothesis about the actual evolutionary history of a group, while "phylogenies" represent true evolutionary history. To other biologists, "cladogram" suggests that the lengths of the branches in the diagram are arbitrary, while in a "phylogeny," the branch lengths indicate the amount of character change. The words "phylogram" and "dendrogram" are also sometimes used to mean the same sort of thing with slight variations. These vocabulary differences are subtle and are not consistently used within the biological community. For our purposes here, the important things to remember are that organisms are related and that we can represent those relationships (and our hypotheses about them) with tree structures.

Evolutionary trees depict clades. A clade is a group of organisms that includes an ancestor and all descendants of that ancestor. You can think of a clade as a branch on the tree of life. Some examples of clades are shown on the tree below.